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Preprint submitted on 18 Jun 2019

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Generic transversality of heteroclinic and homoclinic

orbits for scalar parabolic equations

Pavel Brunovský, Romain Joly, Geneviève Raugel

To cite this version:

Pavel Brunovský, Romain Joly, Geneviève Raugel. Generic transversality of heteroclinic and homo-clinic orbits for scalar parabolic equations. 2019. �hal-02159504�

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Generic transversality of heteroclinic and

homoclinic orbits for scalar parabolic

equations

Pavel Brunovsk´

y

, Romain Joly

and Genevi`

eve Raugel

June 2019

Abstract

In this paper, we consider the scalar reaction-diffusion equations Btu “ ∆u ` f px, u, ∇uq

on a bounded domain Ω Ă Rdof class C2,γ. We show that the heteroclinic and

homoclinic orbits connecting hyperbolic equilibria and hyperbolic periodic or-bits are transverse, generically with respect to f . One of the main ingredients of the proof is an accurate study of the singular nodal set of solutions of linear parabolic equations. Our main result is a first step for proving the genericity of Kupka-Smale property, the generic hyperbolicity of periodic orbits remain-ing unproved.

Key words: transversality, parabolic PDE, Kupka-Smale property, singular nodal set, unique continuation.

2010 AMS subject classification: Primary 35B10, 35B30, 35K57, 37D05, 37D15, 37L45; Secondary 35B40

1

Introduction

Let d ě 2 and let Ω Ă Rd be a bounded domain of class C2,γ, where 0 ă γ ď 1. Let p ą d be fixed, let X “ Lp

pΩq and let

∆D : Dp´∆Dq “ W01,ppΩq X W2,ppΩq ÝÑ X “ LppΩq

Department of Applied Mathematics and Statistics, Comenius University Bratislava, Bratislava 84248, Slovakia.

Universit´e Grenoble Alpes, CNRS, Institut Fourier, F-38000 Grenoble, France, email: romain.joly@univ-grenoble-alpes.fr

Universit´e Paris-Sud & CNRS, Laboratoire de Math´ematiques d’Orsay, 91405 Orsay cedex, France.

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be the Laplacian operator with homogeneous Dirichlet boundary conditions. Let α P p1{2 ` d{2p, 1q, so that Xα “ Dpp´∆Dqαq ãÑ W2α,ppΩq is compactly embedded

in C1

pΩq.

We consider the scalar parabolic equation $

& %

Btupx, tq “ ∆Dupx, tq ` f px, upx, tq, ∇upx, tqq, px, tq P Ω ˆ p0, `8q

upx, tq “ 0, px, tq P BΩ ˆ p0, `8q upx, 0q “ u0pxq P Xα,

(1.1)

where f P C2

pΩ ˆ R ˆ Rd, Rq and upx, tq P R.

The local existence and uniqueness of classical solutions uptq P C0pr0, τ q, Xαq of

Equation (1.1), as well as the continuous dependence of the solutions with respect to the initial data u0 in Xα, are well known (see [31] for example and Section 2

for more details). Thus, Eq. (1.1) generates a local dynamical system Sptq ” Sfptq on Xα. This dynamical system contains all the features of a classical

finite-dimensional system: equilibrium points and periodic orbits, stable and unstable manifolds. . . We recall the definition of these objects, the definition of hyperbolicity and of transversality in Section 3. There, we also present their construction in our framework. Notice that the realizations results of [14] and [53] show the possible existence of very complicated dynamics for (1.1), such as chaotic dynamics, as soon as d ě 2.

In what follows, for any r ě 2, we denote by Cr the space Cr

pΩ ˆ R ˆ Rd, Rq endowed with the Whitney topology, which is a Baire space (see Appendix A for definitions, including the one of generic subset). In fact, our result still holds if we embed Cr with another reasonable topology, but the Whitney one is the most

classical. See [19] and Appendix A below for more details. Our main result is as follows.

Theorem 1.1. Generic transversality of connecting orbits

Let r ě 2 and let f0 P Cr. Let C0´ and C0` be two critical elements of the flow of

(1.1), i.e. C˘

0 are equilibrium points or periodic orbits, C ´ 0 “ C

`

0 being possible.

Assume that both C´

0 and C `

0 are hyperbolic. Then, there exists a neighborhood

O of f0 in Cr and a generic set G Ă O such that:

i) there exist two families C´

pf q and C`pf q of critical elements (either equilibrium points or periodic orbits) of the flow of (1.1), depending smoothly of f P O, such that C˘pf

0q “ C0˘ and C˘pf q is hyperbolic for any f P O.

ii) for any f in the generic set G Ă O, the unstable manifold WupC´pf qq and the

stable manifold WspC`pf qq intersect transversally, i.e. WupC´pf qq&WspC`pf qq. Theorem 1.1 states the generic transversality of connecting orbits, i.e. hetero-clinic and homohetero-clinic orbits, between hpyerbolic critical elements (either equilibrium points or periodic orbits). See Figure 1 for an illustration of a typical transversal connecting orbit. This is a first step to obtain the genericity of Kupka-Smale prop-erty. Below in this introduction, we recall the historical background and previous results. We discuss about the missing ingredients to obtain the genericity of the whole Kupka-Smale property in Appendix C.

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C

´

uptq

W

s

pC

´

q

C

`

W

s

pC

`

q

W

u

pC

´

q

W

u

pC

`

q

Figure 1: A typical transversal heteroclinic orbit connecting a periodic orbit C´

and an equilibrium point C`. If C˘ are hyperbolic, they admit stable and unstable

manifolds. Theorem 1.1 states that, the transversality of uptq in this picture is a generic situation in the parabolic equation (1.1). Here C´ is a periodic orbit

and C` is an equilibrium point. This situation is robust to perturbation and yields

several important qualitative properties of the dynamics. See the third part of this introduction for the historical background and Section 3 for precise definitions.

Notice that we do not need to assume global existence of solutions in Theorem 1.1. Indeed, we consider closed and connecting orbits, which are by definition solu-tions uptq P Xα of (1.1), which are defined for any time t P R and are also uniformly bounded for t P R. So, we do not really care about solutions of Eq. (1.1), which do not exist globally. If one wants that all solutions of (1.1) exist for 0 ď t ď 8, one has to introduce additional hypotheses on f (see [55] for instance).

We also enhance that our result may apply to settings different from (1.1). Typ-ically, we can choose different boundary conditions or consider systems of parabolic equations. We discuss this kind of straightforward generalizations in Section 7. Observability of trajectories, unique continuation and singular nodal sets. As in the classical case of generic transversality in ODEs, the proof of Theorem 1.1 consists in finding suitable perturbation of the linearity f for breaking the non-transversal orbits. Of course, even if the general patterns and the spirit of the proofs stay the same, working with PDE’s instead of ODE’s gives rise to several more or less delicate technical problems. For example, for proving generic properties, instead of using Thom’s transversality theorem (as in [51]), we will apply a Sard-Smale theorem stated in Appendix B. Here, we want to emphasize that, in the case of PDE’s, the main new difficulty arises in the construction of appropriate perturbations. When one wants to prove that a property is dense in the set of ODE’s of the form 9yptq “ gpyptqq, for each g, one has to construct a particular perturbation εh with small ε such that the flow of 9yptq “ pg ` εhqpyptqq satisfies the desired property. The vector field h of the perturbation can be chosen freely and localized, so that his support intersects the trajectory of yptq only in the neighborhood of ypt0q. In the case of PDE’s, we have to construct a perturbation h of the

non-linearity such that the flow of Btupx, tq “ ∆upx, tq ` pf ` εhqpx, upx, tq, ∇upx, tqq

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form

up¨q P Xα ÞÝÑ hp¨, up¨q, ∇up¨qq (1.2) Since two distinct functions u1 and u2 can take the same value pu1px0q, ∇u1px0qq “

pu2px0q, ∇u2px0qq at a given x0 P Ω, the perturbations of the form (1.2) are in

general “non local” in Xα. Given a particular trajectory uptq and a time t 0, our

strategy consists in constructing a perturbation (1.2), whose support, even if it is large, intersects upx, tq only around px0, t0q, which allows to consider (1.2) as a local

perturbation. However, this construction is not straightforward and requires deep properties of the PDE. This problem is close to observability questions: how much information on a solution uptq can we get from the observation at one point x0 of

upx0, tq and ∇upx0, tq?

To be able to prove Theorem 1.1, we will prove in Section 5 results of the following type.

Theorem 1.2. Injectivity properties of connecting orbits Let f P C8

pΩ ˆ R ˆ Rd, Rq. Let uptq be a heteroclinic or homoclinic orbit connecting two critical elements. Then there exists a dense open set of points px0, t0q P Ω ˆ R

such that the curve t ÞÑ pupx0, tq, ∇upx0, tqq is one to one at t0 in the sense that:

i) pBtupx0, t0q, ∇Btupx0, t0qq ‰ 0,

ii) for all t P R, pupx0, tq, ∇px0, tqq “ pupx0, t0q, ∇px0, t0qq ùñ t “ t0.

The above result is a key property to be able to construct a suitable perturbation of the non-linearity f in the proof of Theorem 1.1. The following result is similar: it shows that the period of a periodic orbit of the parabolic equation may be observed very locally. This result is not required in the proof of our main theorem, but it may be interesting by itself and could be a key step to prove the generic hyperbolicity of periodic orbits (see the discussion of Appendix C).

Theorem 1.3. Pointwise observability of the period of periodic orbits Let f P C8

pΩ ˆ R ˆ Rd, Rq. Let pptq be a periodic solution of (1.1) with minimal period ω ą 0. Then there exists a dense open set of points px0, t0q P Ω ˆ R such that

pppx0, tq, ∇ppx0, tqq “ pppx0, t0q, ∇ppx0, t0qq ùñ t P t0 ` Zω .

Notice that in dimension d “ 1, the above results are true for all px0, t0q and not

only for a dense subset (see [37]).

To obtain these injectivity properties of px, tq ÞÝÑ px, upx, tq, ∇upx, tqq, where uptq “ Sfptqu0 is a bounded complete trajectory of (1.1), we set

vpx, t, τ q “ upx, tq ´ upx, t ` τ q ,

and remark by using the equation (1.1) that vpx, tq is the solution of a linear parabolic equation with parameter of the form

Btvpx, t, τ q “ ∆vpx, t, τ q ` apx, t, τ qvpx, t, τ q ` bpx, t, τ q.∇xvpx, t, τ q , (1.3)

in the domain Ω of Rd. The non-injectivity points of the image of px, upx, tq, ∇upx, tqq,

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points px, t, τ q where vpx, t, τ q and ∇xvpx, t, τ q both vanish. The singular nodal set

of solutions of the parabolic equations, with coefficients independent of the param-eter τ , have already been studied in [28] and in [10] for example. Here, generalizing an argument of [29] and applying unique continuations results (recalled in Section 2), we prove the following theorem, see Section 4.

Theorem 1.4. Singular nodal sets for parabolic PDEs with parameter Let I and J be open intervals of R. Let a P C8

pΩˆI ˆJ, Rq and b P C8pΩˆI ˆJ, Rdq be bounded coefficients. Let v be a strong solution of (1.3) with Dirichlet boundary conditions. Let r ě 1 and assume that v is of class Cr with respect to τ and of class C8 with respect to x and t. Assume moreover that the null solution is not part

of the family, that is that, there are no time t P I and parameter τ P J such that vp., t, τ q ” 0.

Then, the set

tpx0, t0q P Ω ˆ I | E τ P J such that pvpx0, t0, τ q, ∇vpx0, t0, τ qq “ p0, 0qu

is generic in Ω ˆ I. In other words, the projection of all the singular nodal sets of the family of solutions vp¨, ¨, τ q is negligible in Ω ˆ I.

Historical background: the Morse-Smale and Kupka-Smale properties. The transversality of unstable and stable manifolds stated in Theorem 1.1 is related to the local stability of the qualitative dynamics. In the modeling of phenomena in physics or biology, we often work on approximate systems: some phenomena are neglected, only approximate values of the parameters are known, or we work with a discretized version of the system for simulation by computer. . . Therefore, it is important to know if such small approximations may qualitatively change the dy-namics or not. Unfortunately, when perturbing general dynamical systems, drastic changes in the local or global dynamics can occur due for example to bifurcation phenomena. Thus, the common hope is that these bifurcations are rare, that is, that the systems, whose dynamics are robust under perturbations, are dense or generic. Here, we obtain the generic transversality of heteroclinic and homoclinic orbits between critical elements. Roughly, Theorem 1.1 says that if we consider two hyperbolic closed orbits of the flow of the parabolic equation (1.1) and if we observe a connecting orbit between them, then, “almost surely” this connection still remains after small perturbations of the system (numerical computation, changes of the parameters. . . ).

Such stability questions have been extensively studied in the case of vector fields or iterations of maps. In 1937, Andronov and Pontrjagin introduced the fundamental notion of structurally stable vectors fields (“syst`emes grossiers” or “coarse systems”), that is, vector fields X0 which have a neighborhood V0 in the C1-topology such that

any vector field X in V0 is topologically equivalent to X0. In 1959 ([63]), Smale

defined the class of nowadays called Morse-Smale dynamical systems on compact n´dimensional manifolds, that is, systems for which the non-wandering set consists only in a finite number of hyperbolic equilibria and hyperbolic periodic orbits and for which the intersections of the stable and unstable manifolds of equilibria and periodic orbits are all transversal. Peixoto ([50]) proved that Morse-Smale vector fields are

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dense and have structurally stable qualitative dynamics in compact orientable two-dimensional manifolds. In 1968, Palis and Smale ([46], [48]) proved the structural stability of the Morse-Smale dynamical systems in any dimension. However, the density of Morse-Smale systems fails in dimension higher than two, due to “Smale horseshoe”. In 1963, Smale ([65]) and also Kupka ([41]) introduced the Kupka-Smale vector fields, that is, the vector fields for which all the equilibria and periodic orbits are hyperbolic and the intersections of the stable and unstable manifolds of equilibria and periodic orbits are all transversal. They both show the density of such systems in any dimension (see also [51]). The qualitative dynamics of Kupka-Smale systems are locally stable: periodic orbits, the local dynamics around them and their connections move smoothly when a parameter of the equation is changing.

For the partial differential equations (PDE’s in short), the history of structural stability and of local stability is more recent. Notice that a trajectory of the dynam-ical system Sptq generated by such a PDE is of the form t ÞÑ Sptqu0 “ up¨, tq, where

upx, tq is the solution of the PDE with initial data u0pxq. In particular, the trajectory

moves in a functions space (often a Sobolev space), which is infinite-dimensional. As a generalization of [46] and [48], [26] and [45] proved that Morse-Smale and Kupka-Smale properties are still meaningful in infinite-dimensional systems for the problem of stability of the qualitative dynamics. Therefore, there is a great inter-est in obtaining generalizations of the above mentioned finite-dimensional generic results. Notice that, if we want to get a meaningful genericity result, we have to allow perturbations only in the same class of PDE’s. Typically, the parameter with respect to which the genericity is obtained is the non-linearity f .

The first example of transversality of unstable and stable manifolds for PDE’s is due to Henry ([30]) in 1985 for the reaction-diffusion equation in the segment

Btu “ uxx` f px, u, uxq, px, tq P p0, 1q ˆ p0, `8q (1.4)

with Dirichlet, Neumann or Robin boundary conditions. More strikingly, he ob-tained the noteworthy property that the stable and unstable manifolds of two hyper-bolic equilibria of (1.4) always intersect transversally. A key ingredient for proving this automatic transversality is the use of the non-increase of the “Sturm number” or “zero number” [69] of the solutions of the corresponding linearized parabolic equa-tions. In addition to this automatic transversality, the gradient structure proved in [72] shows the genericity of Morse-Smale property for the flow of (1.4) with separated boundary conditions.

If we consider (1.4) with periodic boundary conditions, that is the parabolic equation on the circle S1

Btu “ uxx` f px, u, uxq, px, tq P S1ˆ p0, `8q (1.5)

then the gradient structure fails but the flow of (1.5) still has particular properties equivalent to the ones of two-dimensional ODEs, such as the Poincar´e-Bendixson property proved in [18] (the reader interested in the correspondence between the dynamics of (1.4) and the ones of low-dimensional ODEs may consider the review paper [39]). In 2008, still using the powerful tool of the “zero number”, Czaja and Rocha ([13]) proved that, for the parabolic equations on the circle (1.5), the stable and unstable manifolds of hyperbolic periodic orbits always intersect transversally.

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In 2010, the second and third authors completed the results of Czaja and Rocha. More precisely, they proved in [37] that the equilibria and periodic orbits are hyper-bolic, generically with respect to the nonlinearity f . They also proved that the stable and unstable manifolds of hyperbolic critical elements C´ and C` intersect

transver-sally, unless both critical elements C´ and C` are equilibria of same Morse index

and moreover that, generically with respect to f , such connecting orbits between equilibria with the same Morse index ([38]) do not exist. Finally, the Poincar´ e-Bendixson theorem of [18] yields that, generically with respect to f , the equation (1.5) is Morse-Smale (see [38]).

Concerning spatial dimension higher than d “ 1, the generic transversality of stable and unstable manifolds has been shown in 1997 by the first author and P. Pol´aˇcik ([7]) in the case f ” f px, uq, that is, for the equation

Btu “ ∆u ` f px, uq, px, tq P Ω ˆ p0, `8q (1.6)

with Ω Ă Rd, d ě 2. As a consequence, since (1.6) is a gradient system, they deduce

that, under additional dissipative conditions on the non-linearity, the Morse-Smale property holds for the flow (1.6) generically with respect to f P C2. It is noteworthy, as shown by Pol´aˇcik ([54]), that this generic transversality property is not true if one considers homogeneous functions f px, uq ” f puq only.

We also mention that generic transversality properties have been shown by the authors for various gradient damped wave equations, see [8] and [36].

Due to the realization results of Dancer and Pol´aˇcik, [14] and [53], we know that the dynamics of the flow of the general parabolic equation (1.1) in dimension d ě 2 may be as complicated as chaotic flows. We may only hope to prove the genericity of the Kupka-Smale property and not of the Morse-Smale one. Notice that the flow of (1.1) is not gradient (periodic orbits may exist) and the very particular and helpful “zero number property” of spatial dimension d “ 1 fails. In the present paper, we prove the generic transversality property. The generic hyperbolicity of equilibrium points is already proved in [37] in any space dimension. Thus, the generic hyperbolicity of periodic orbits is the only remaining step to obtain the genericity of the Kupka-Smale property.

Some years ago, in a preliminary draft of this paper, we were convinced to have proved the genericity of the Kupka-Smale property. However, Maxime Percy du Sert pointed to us a gap in the proof of generic hyperbolicity of periodic orbits. We did not manage to fill it. Recently, two of the three authors passed away and we decided to publish the results as obtained together. In particular, we prove the generic transversality only (unlike claimed in [39]). In Appendix C, we quickly discuss our ideas to obtain the generic hyperbolicity of periodic orbits and indicate where the gap remains.

Plan of the article.

In Section 2, we recall the classical existence and uniqueness properties of the solu-tions of the scalar parabolic equation and the corresponding linear and linear adjoint equations. We also review unique continuation properties, which are fundamental in this paper. In Section 3, we remind some basic definitions such as hyperbolicity of critical elements and we state the main properties of the dynamical system Sfptq,

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namely the existence of C1 immersed finite-codimensional (resp. finite-dimensional)

stable (resp. unstable) manifolds of hyperbolic critical elements. Section 4 is devoted to the study of the singular nodal sets and to the proof of Theorem 1.4. In Section 5, we show that Theorem 1.4 leads to one-to-one properties such as Theorems 1.2 and 1.3. Using these tools, in Section 6, we prove Theorem 1.1, i.e. we show the generic transversality of heteroclinic and homoclinic orbits of the parabolic equation (1.1). Section 7 contains discussions about some generalizations of Theorem 1.1. We con-clude by two appendices recalling the basic facts about the Whitney topology and Sard-Smale theorems, which will be used in this paper, and one appendix discussing the still open problem of generic hyperbolicity of periodic orbits of (1.1).

Dedication: Very sadly, both Pavol Brunovsk´y and Genevi`eve Raugel passed away before the publication of this article, respectively in december 2018 and in may 2019. They were still working actively on the manuscript and the present version is exactly the one which have been completed by them. This article is dedicated to their memories.

Acknowledgement: The last two authors have been funded by the research project ISDEEC ANR-16-CE40-0013.

2

Some basic results on parabolic PDEs

2.1

Local existence and regularity results of the parabolic

equation (1.1)

The solutions of the scalar parabolic equation (1.1) exist locally and are unique, see for example [49] or [31]. In the whole paper, α belongs to the open interval p12`2pd, 1q. We recall that we use the notation f P Cr

pE, Rq to indicate the regularity of f , i.e. to say that the function f : E Ñ R is of class Cr. Where a topology is required (smooth dependences on f etc.), the notation Cr

pE, Rq refers to the space CrpE, Rq endowed with the Whitney topology (see Appendix A).

Proposition 2.1. Let r ě 1 and f P Cr

pΩ ˆ R ˆ Rd, Rq.

i) For any u0 P Xα, there exists a maximal time T pu0q ą 0 such that (1.1) has

a unique classical solution Sfptqu0 “ uptq P C0pr0, T s, Xαq X C1pp0, T s, Xβq X

C0

pp0, T s, Dp´∆Dqq, for any 0 ď β ă 1 and for any 0 ă T ď T pu0q. If T pu0q is

finite, then }uptq}Xα goes to `8 when t ă T pu0q tends to T pu0q.

Moreover, t ÞÑ Btuptq is locally H¨older continuous from p0, T s into Xβ, for

0 ď β ă 1. In particular, up¨q ” Sfp¨qu0 belongs to the space C0pp0, T s, W3,ppΩqqX

C1pp0, T s, Ws,ppΩqq, for any s ă 2, and thus belongs to the spaces C0pp0, T s, C2pΩqqX

C1

pp0, T s, C1pΩqq and C1pΩˆrτ, T s, Rq, for any 0 ă τ ă T . If, in addition, the first derivatives Duf px, ¨, ¨q and D∇uf px, ¨, ¨q are Lipschitz-continuous on the bounded

sets of Ω ˆ R ˆ Rd, then up¨q belongs to C1pp0, T s, W2,ppΩqq X C2pp0, T s, Ws,ppΩqq,

for any s ă 2 and hence up¨q also belongs to C2pΩ ˆ rτ, T s, Rq, for any 0 ă τ ă T . ii) For any u0 P Xα, for any T ă T pu0q, there exist a neighborhood U ” U pT q

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and any g P V, vptq ” Sgptqv0 is well defined on r0, T s, depends continuously on

v0 P Xα and g P C1, and there exists a positive number R ” RpT, U , Vq such that

ppSgptqv0qpxq, p∇Sgptqv0qpxqq belongs to the ball BRd`1p0, Rq, for all pt, v0, g, xq P

r0, T s ˆ U ˆ V ˆ Ω.

iii) Moreover, for any u0 P Xα, for any T ă T pu0q, the map pt, u0q P p0, T s ˆ U ÞÑ

Sfptqu0 P Xα is of class Cr and, in particular, Sfptq is a local semigroup of

class Cr. In addition, there exists a neighborhood W of f in the space Cr

pΩ ˆ r´2R, 2Rs ˆ r´2R, 2Rsd, Rq such that the map pt, u0, gq P p0, T s ˆ U ˆ W ÞÑ

Sgptqu0 P Xα is of class Cr.

Remarks:

1) The statement (i) is a direct consequence of the existence and regularity re-sults given in [31, Chapter 3] and of elliptic regularity properties. We only want to emphasize that, since the solution up¨q ” Sfp¨qu0 belongs to C0pr0, T s, Xαq

and that Xα is continuously embedded in C1

pΩq, up¨q automatically belongs to the space C0pr0, T q, C1pΩqq. Since up¨q is a classical solution and belongs

to C0pp0, T s, W2,ppΩqq X C1pp0, T s, W1,ppΩqq, f px, u, ∇uq ´ Btu is in the space

C0

pp0, T s, W1,ppΩqq and the regularity properties of the elliptic equation ∆Du “ Btu ´ f px, u, ∇uq ,

imply that up¨q belongs to the space C0

pp0, T s, W3,ppΩqq Ă C0pp0, T s, C2pΩqq. 2) Statements (ii) and (iii) are also easy consequences of [31, Theorem 3.4.4 and

Corollary 3.4.5]. We want to point out that, for any u0 P Xα and any 0 ă

T ă T pu0q, there exists R0 ą 0 such that pupx, tq, ∇upx, tqq, for all px, tq P

Ω ˆ r0, T s is bounded in Rd`1 by a positive number R

0 ” R0pu0, T q. Since

gpx, upx, tq, ∇upx, tqq depends only on the values of x, upx, tq and ∇upx, tq, we can show, by applying the continuity results of [31, Section 3.4], that, for any R ą R0, for any 0 ă ε ă pR ´ R0q{2, there exists a positive number η such that,

for any gp¨, ¨, ¨q P CrpΩ ˆ r´R, Rs ˆ r´R, Rsd, Rq, η-close to f in the classical norm

of CrpΩ ˆ r´R, Rs ˆ r´R, Rsd, Rq, ppSgptqu0qpxq, p∇Sgptqu0qpxqq belongs to the

ball BRd`1p0, R0` εq, for all px, tq P Ω ˆ r0, T s.

3) Notice that the statement (ii) of Proposition 2.1 implies that the maximal time T pu0q is a lower-semi-continuous function of the initial data u0

As we have already seen, the parabolic equation has a smoothing effect at any finite positive time. If the boundary of the domain Ω was of class C8and f belonged

to C8

pΩ ˆ R ˆ Rd, Rq, the solutions of Eq.(1.1) would be in C8pΩ ˆ rτ, T s, Rq for any 0 ă τ ă T ă T pu0q. However, if f P C8pΩ ˆ R ˆ Rd, Rq, we can still show that

the solutions are regular in the interior of Ω, even if Ω is of class C2,α only.

In the whole paper, we say that uptq : t P R ÞÑ uptq is a bounded complete solution (or trajectory) of (1.1) if it is a solution of (1.1), defined for any t P R and bounded in Xα, uniformly with respect to t P R.

Since we are only interested in the regularity of the bounded complete solutions of (1.1), we will state a C8-regularity result for such solutions.

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Proposition 2.2. Assume that f belongs to C8

pΩ ˆ R ˆ Rd, Rq. Then, any bounded complete solution uptq of (1.1) belongs to C8

pΩ ˆ R, Rq. More precisely, for any open set O, such that O Ă Ω, for any R ą 0, any m P N, any k P N, and any q P r1, 8s, there exists a positive constant KpO, R, m, k, qq, such that any bounded complete solution uptq, with suptPR }uptq}Xα ď R, satisfies

sup tPR › › › › dku dtkptq › › › › Wm,qpOq ď KpO, R, m, k, qq . (2.1)

Proof: We will not give all the details of the proof, but will indicate only the main arguments. The proof consists in a recursion argument with respect to k and m. Let uptq be a bounded complete solution of (1.1) satisfying suptPR}uptq}Xα ď R.

First step: Since f belongs to C8

pΩˆRˆRd, Rq, by [31, Corollary 3.4.6], the function t P R ÞÑ uptq P Xα is of class Ck, for any k P N and dku

dtkptq P C0pR, XαX W2,ppΩqq X

C1

pR, Xβq, for any β ă 1, is a classical solution of the equation d dtp dku dtkq “ ∆ dku dtk ` dk dtkpf px, u, ∇uqq . (2.2)

We notice that the term dtdkkpf px, u, ∇uqq can be computed by using the Faa Di

Bruno formula [16] and its generalization [9] as follows. We introduce the pd ` 1q-dimensional vector wpx, tq “ pu, ∇uqpx, tq, that is w1 “ u and wi`1 “ Bxiu. Using

the generalized Faa Di Bruno formula ([9]), we can write, dk dtkpf px, upx, tq,∇upx, tqqq “ ÿ mj“1,|m|“1 Dmwf px, wpx, tqqd k dtkpwjqpx, tq ` ÿ 2ď|m|ďk Dmwf px, wpx, tqq ÿ ppk,mq k!Πkj“1d`j dt`jw ‰nj pnj!qr`j!s|nj| ” ÿ mj“1,|m|“1 Dwmf px, wpx, tqqd k dtkpwjqpx, tq ` gkpx, tq (2.3) where ppk, mq “ tpn1, . . . , nk; `1, . . . , `kq | Ds P v1, kw, ni “ `i “ 0 for 1 ď i ď n ´ su

and gk contains only derivatives with respect to t of order less or equal to k ´ 1.

We notice that the estimate (2.1) for k “ 0, m “ 2 and q “ p is a direct consequence of the hypothesis and of Proposition 2.1. Using (2.3), the fact that W1,p

pΩq is an algebra and the bound suptPR}uptq}Xα ď R, one shows by recursion

on k that sup tPR }d ku dtkptq}W2,ppΩq ď C2pR, kq , (2.4)

where C2pR, kq is a positive constant depending only on R, k (and of f ). Like in

the remarks following Proposition 2.1, the elliptic regularity properties allow also to deduce from Eq.(2.2) and from the estimate (2.4) that,

sup

tPR

}d

ku

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where C3pR, kq is a positive constant depending only on R, k (and of f ).

Second step: One easily shows, by recursion on n P N (and also k) that,

sup

tPR

}d

ku

dtkptq}W3`n,ppOq ď C3`npO, R, kq . (2.6)

Indeed, let Oj, j “ 1, 2, . . . , n ` 1, be a sequence of regular open sets such that

O Ă On`1 Ă On`1 Ă On Ă . . . Ă Oj`1 Ă Oj`1 Ă Oj. . . Ă O1 Ă O1 Ă Ω and ϕj,

j “ 1, 2, . . ., be a corresponding sequence of regular functions such that ϕjpxq P r0, 1s,

x P Ω, and ϕjpxq ” 0, for x P ΩzOj and ϕjpxq ” 1, for x P Oj`1. We recall that, by

the remarks following Proposition 2.1, one already knows that the estimates (2.5) hold for any k P N. We remark that ϕ1u is a solution of the elliptic equation

∆pϕ1uq “ ϕ1

du

dt ` u∆ϕ1` 2∇u ¨ ∇ϕ1´ ϕ1f px, u, ∇uq (2.7) where ϕ1dudt ` u∆ϕ1` 2∇u ¨ ∇ϕ1´ ϕ1f px, u, ∇uq belongs to W3´1,ppO1q X W01,ppO1q.

By the elliptic regularity results, ϕ1u belongs to W3`1,ppO1q and

sup

tPR

}ϕ1uptq}W3`1,ppO

1q ď C3`1pO1, R, 0, ϕ1q , (2.8)

where C3`1pO1, R, 0, ϕ1q is a positive constant depending only on O1, R, ϕ1.

Like-wise, writing the elliptic equality satisfied by ∆pϕ1pd

k

dtkuqq and using the equalities

(2.2) and (2.3), one shows, by recursion on k, that dtdkkpϕ1uq belongs to W

3`1,p pO1q and sup tPR }d k dtkpϕ1uqptq}W3`1,ppO1q ď C3`1pO1, R, k, ϕ1q , (2.9)

where C3`1pO1, R, k, ϕ1q is a positive constant depending only on O1, R, k and ϕ1.

We notice that dtdkkpϕ1uqpxq “ d

k

dtkupxq, for any x P O2.

We next assume that dtdkkpϕjuq belongs to W

3`j,p

pOjq and that the estimates

(2.8) and (2.9) hold with 1 replaced by j. Remarking that ϕj`1u is a solution of the

elliptic equation

∆pϕj`1uq “ ϕj`1

du

dt ` u∆ϕj`1` 2∇u ¨ ∇ϕj`1´ ϕj`1f px, u, ∇uq (2.10) where ϕj`1dudt`u∆ϕj`1`2∇u¨∇ϕj`1´ϕj`1f px, u, ∇uq belongs to W3`j´1,ppOj`1qX

W01,ppOj`1q, we at once show that ϕj`1u belongs to W3`j`1,ppOj`1qXW

1,p

0 pOj`1qand that

the estimate (2.8) holds with 1 replaced by j ` 1. Likewise, one shows by recursion on k that dk

dtkpϕj`1uq belongs to W

3`j`1,p

pOj`1q and that the estimate (2.9) holds

with 1 replaced by j ` 1. Thus, we have proved by recursion on n and k that dtdkkpuq

belongs to W3`n,p

pOq and that the estimates (2.6) are satisfied.

The general estimate (2.1) is a direct consequence of the estimates (2.6) and the classical Sobolev embedding theorem. ˝

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2.2

The linear and linear adjoint equations

Let 0 ď s ă T and let ap¨q P C1

pr0, T s, L8pΩqq and bp¨q P C1pr0, T s, W1,8pΩqdq. We consider solutions v of the linear parabolic equation

vtpx, tq “∆Dvpx, tq ` apx, tqvpx, tq ` bpx, tq.∇vpx, tq , t ą s, x P Ω,

vpx, sq “vs .

(2.11)

In what follows, we denote Aptq the operator

Aptq “ ∆D` apx, tq. ` bpx, tq.∇ .

Equation (2.11) arises either when one linearizes the parabolic equation (1.1) along a solution u, in which case we have

" apx, tq “ f1

upx, upx, tq, ∇upx, tqq

bpx, tq “ f1

∇upx, upx, tq, ∇upx, tqq

(2.12)

or when one considers the difference vptq “ u2ptq ´ u1ptq between two solutions u1

and u2 of (1.1), in which case we have

#

apx, tq “ş01f1

upx, pθu2` p1 ´ θqu1qpx, tq, ∇pθu2` p1 ´ θqu1qpx, tqqdθ

bpx, tq “ş01f1

∇upx, pθu2 ` p1 ´ θqu1qpx, tq, ∇pθu2` p1 ´ θqu1qpx, tqqdθ

(2.13)

Notice that, since f belongs to C2

pΩ ˆ R ˆ Rd, Rq, due to Proposition 2.1, in both cases the coefficients of (2.11) belong to C1pp0, T s, W1,8pΩqq. Since in what follows,

we are mainly applying the results of this section to bounded complete trajectories, we can consider, without loss of generality, that the coefficients of (2.11) belong to C1pr0, T s, W1,8pΩqq.

Proposition 2.3. Let r P r1, 8q and let vs P LrpΩq. Equation (2.11) has a unique

solution vptq ” U pt, sqvs P C0prs, T s, LrpΩqq X C1pps, T s, LrpΩqq X C0pps, T s, W2,rpΩq X

W01,rpΩqq satisfying vpsq “ vs. Moreover, v : t P ps, T s ÞÑ vptq P Xα is H¨older

continuous and belongs to C1pps, T s, LqpΩqq X C0pps, T s, W2,qpΩq X W01,qpΩqq for any

q P r1, `8s. In particular v P C0pps, T s, C1pΩqq.

Proof: For the existence, uniqueness and regularity of the solution of vptq ” U pt, sqvs in C0prs, T s, LrpΩqq X C1pps, T s, LrpΩqq X C0pps, T s, W2,rpΩq X W01,rpΩqq, we

refer to [31, Theorem 7.1.3]. To prove that vptq belongs to any space LqpΩq (and

thus to Xα), we will use a bootstrap argument. Assume that vs belongs to LrpΩq

and set r “ r0. By [31, Theorem 7.1.3], vps ` δq P W2,r0pΩq for any δ ą 0. If

d ´ 2r0 ď 0, then, vps ` δq P W2,r0pΩq Ă LqpΩq, for any positive number q ě 1, by

the classical Sobolev embedding. If, d ´ 2r0 ą 0, again by the Sobolev embedding

theorem, vps ` δq P W2,r0pΩq Ă Lr1pΩq, for r

1 “ dr0{pd ´ 2r0q “ r0` 2r02{pd ´ 2r0q.

We again apply [31, Theorem 7.1.3] to deduce that vps ` 2δq P W2,r1pΩq, for any

δ ą 0. Again, if d ´ 2r1 ą 0, we obtain that vpt ` 2δq P W2,r2pΩq Ă Lr2pΩq, for

r2 “ dr1{pd ´ 2r1q ě r1` 2r12{pd ´ 2r1q ě r0` 2r02{pd ´ 2r0q ` 2r21{pd ´ 2r1q. Clearly,

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number of steps, we obtain that vptq P LqpΩq. ˝

Proposition 2.3 tells that Equation (2.11) generates a family of evolution opera-tors U pt, sq on LppΩq, which is extended to LrpΩq for any r ě 1.

Let now 1 ă p ă `8, which implies that X “ LppΩq is reflexive. Denote by p˚ the conjugate exponent of p, that is, p˚

“ p{pp ´ 1q; consider the adjoint space X˚ “ pLppΩqq˚ “ Lp˚pΩq of X and the adjoint evolution operator U pt, sq˚ : X˚ Ñ X˚. Let

T ą 0; for ψT P Lp

˚

pΩq, we define the function ψ : s P r0, T s ÞÑ ψpsq “ U pT , sq˚ψT.

In general, ψpsq is only a weak˚ solution of the equation

Bsψpx, sq “ ´∆Dψpx, sq ´ apx, sqψpx, sq ` divpbpx, sqψpx, sqq (2.14)

with px, sq P Ω ˆ p0, T q and with final data ψpT q “ ψT in the weak-˚ sense. More

precisely, s P r0, T q ÞÑ ψpsq P X˚ is locally H¨older continuous, for each φ P X,

xφ, ψpsqy Ñ xφ, ψTy when s Ñ T´and, for each φ P DpA˚q, pφ, ψpsqq is differentiable

on r0, T q with Btpφ, ψpsqq “ pApsqφ, ψpsqq.

Usually, ψpsq “ U pT, sq˚ψ

T is only a solution of (2.14) in a weak sense. But

here, since ap¨q P C1pr0, T s, L8pΩqq and bp¨q P C1pr0, T s, W1,8pΩqdq, ψpsq is a strong

solution of (2.14), as we shall see in the proposition below. Notice that (2.14) is a parabolic equation solved backwards in time.

Proposition 2.4.

1) With the above notations, ψpsq “ U pT, sq˚ψ

T belongs to C1pr0, T q, X˚qXC0pr0, T q,

W2,p˚

pΩq X W1,p

˚

0 pΩqq. Moreover, it satisfies (2.14) in the strong sense and ψpsq

belongs to C1pr0, T q, LqpΩqq X C0pr0, T q, W2,qpΩq X W1,q

0 pΩqq for any q ě 1.

2) Let ˜ψT P X˚. For any 0 ă η ă T , ˜ψT ´η “ U pT , T ´ ηq˚pp´∆Dqαq˚ψ˜T

is well defined in X˚. Hence, for s ă T ´ η, ˜ψpsq “ U pT ´ η, sq˚ψ˜ T ´η “

U pT, sq˚pp´∆

Dqαq˚ψ˜T belongs to C1pr0, T ´ ηq, X˚q X C0pr0, T ´ ηq, W2,p

˚

pΩq X W01,p˚pΩqq and a strong solution of (2.14).

Proof: The first part of the proposition is a direct consequence of [31, Theorem 7.3.1] on the existence and regularity of solutions for the adjoint equation and on the fact that the coefficients have the regularity ap¨q P C1pr0, T s, L8pΩqq and bp¨q P C1

pr0, T s, W1,8pΩqdq. The fact that ψpsq belongs to any LqpΩq is proved by recursion as in Proposition 2.3.

To show the second part of the proposition, let ˜ψT P X˚ and let ϕ P X “

Lp

pΩq. By Proposition 2.3, U pT , T ´ ηqϕ belongs to Xα “ Dpp´∆Dqαq and thus

x ˜ψT|p´∆DqαU pT, T ´ηqϕyLp˚,Lpis well defined. Therefore, U pT, T ´ηq˚pp´∆Dqαq˚ψ˜T

is well defined and belongs to Lp˚

pΩq. To finish, we apply [31, Theorem 7.3.1] (or the first part of the proposition) to the initial data ψT “ U pT , T ´ ηq˚pp´∆Dqαq˚ψ˜T.

˝

2.3

Unique continuation properties

In this section, we recall some important unique continuation properties satisfied by the linear parabolic equation (2.11). We enhance that these properties will apply to

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solutions vptq P Xα of (2.11) with coefficients given by (2.12) or (2.13). Hence, we

may apply it to the difference of two solutions of the nonlinear parabolic equation (1.1). In particular, the unique continuation properties below will have fundamental consequences on the properties of the dynamics of (1.1), such as the injectivity of the flow.

The following result is a direct consequence of the backward uniqueness property stated in [4, Theorem II.1].

Proposition 2.5.

1) Let T ą 0. Let apx, tq P L8

pΩ ˆ p0, T qq and let bpx, tq P L8pΩ ˆ p0, T qqd. Let vptq P L2

pp0, T q, H01pΩqq be a solution of the linear parabolic equation (2.11).

Then, vpT q ” 0 in Ω if and only if v vanishes identically in p0, T q ˆ Ω.

2) Likewise, assume that apx, tq P L8pΩ ˆ p0, T qq, that bpx, tq P L8pΩ ˆ p0, T qqd

and that Dxibpx, tq P L

8

pΩ ˆ p0, T qqd, 0 ď i ď d. Let ψptq P L2pp0, T q, H01pΩqq be

a solution of the adjoint linear equation (2.14). Then, ψp0q ” 0 in Ω if and only if ψ vanishes identically in p0, T q ˆ Ω.

Let now u1 and u2 be two solutions on the time interval r0, T s of the equation

(1.1). We already remarked that vptq “ u2ptq ´ u1ptq satisfies the linear equation

(2.11) with the coefficients a and b given by (2.13). By Proposition 2.1, the coef-ficients a, b and the function vptq satisfy the regularity assumptions of the above proposition 2.5. Thus, if u1pT q “ u2pT q, then u1 ” u2 on r0, T s. This leads to state

the following corollary.

Corollary 2.6. Let T ą 0. Let u1ptq and u2ptq be two solutions on the time interval

r0, T s of the equation (1.1). If u1pT q “ u2pT q, then u1ptq “ u2ptq, for any t P r0, T s.

In other terms, the local dynamical system Sfptq generated by (1.1) has the backward

uniqueness property.

The following result is proved in [62] and shows that the set of the zeros of the solutions of the linear parabolic equation is a closed set with empty interior.

Proposition 2.7. Let T ą 0, a and b be as in Proposition 2.5. We assume that vpx, tq P L2pp0, T q, H2pΩq X H1

0pΩqq is a solution of the linear parabolic equation

(2.11). If vpx, tq vanishes on an open non-empty subset of Ω ˆ p0, T q, then vpx, tq identically vanishes on Ω ˆ p0, T q.

A similar result has been obtained for the strong solutions of the adjoint equation in [17, Corollary 2.12].

Proposition 2.8. Let T ą 0. Let apx, sq P L8

pΩ ˆ p0, T qq and let bpx, sq P L8pΩ ˆ p0, T qqd. Let ψpsq P L2pp0, T q, H2pΩq X H1

0pΩqq be a solution of the adjoint equation

(2.14). If ψpx, tq vanishes on an open non-empty subset of Ω ˆ p0, T q, then ψpx, tq identically vanishes on Ω ˆ p0, T q.

In the particular case of smooth solutions of (2.11) (typically if one considers global bounded solutions and a smooth non-linearity f ), we will need stronger prop-erties on the zeros of the solutions in Section 4.

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We say that v vanishes to infinite order in both the space and time variables at px0, t0q if, for any k ě 1, there is a constant Ck ą 0, such that, for any px, tq P

Ω ˆ r´T, 0s,

|vpx, tq| ď Ckp|x ´ x0|2` |t ´ t0|qk{2 . (2.15)

We shall often apply the following unique continuation result of Escauriaza and Fern´andez [15].

Proposition 2.9. Assume that v P C0pp´T , 0s, C2pΩqqXC1pp´T , 0s, C1pΩqq is a

solu-tion of (2.11) and satisfies either homogeneous Dirichlet or homogeneous Neumann boundary conditions. Suppose that v vanishes to infinite order at px0, 0q in both the

space and time variables in the sense of (2.15). Assume moreover that there exists a positive constant K such that for any px, tq P Ω ˆ p´T, 0s,

|vtpx, tq ´ ∆vpx, tq| ď Kp|∇vpx, tq| ` |vpx, tq|q . (2.16)

Then, vpx, 0q vanishes for any x P Ω and therefore vpx, tq identically vanishes in Ω ˆ r´T, 0s.

We say that v vanishes to infinite order in space at px0, t0q if, for any k ě 1,

there is a constant Ck ą 0, such that

|vpx, t0q| ď Ck|x ´ x0|k . (2.17)

From Proposition 2.9 and [2, Theorem 1], we deduce the following unique continua-tion result for solucontinua-tions v P C0pp´T , 0s, C2pΩqq X C1pp´T , 0s, C1pΩqq of (2.16), which

vanish to infinite order in space. The following result can also be deduced from Proposition 2.9, a simple computation and, a recursion argument when vpx, tq is a C8-function in the variables px, tq. Indeed, if for example vpx, t

0q vanishes to order

2 (resp. 4) in space at px0, t0q, then, due to the equation (2.11), vtpx, t0q vanishes

to order 0 (resp. 2) in space at px0, t0q. Moreover, if vpx, t0q vanishes to order 4

in space at px0, t0q, deriving the equation (2.11) with respect to t, one shows that

vttpx, tq vanishes at order 0 in space. Finally, continuing the recursion argument on

k and on the derivatives with respect to t, one shows that v vanishes to infinite order at px0, t0q in both the space and time variables in the sense of (2.15)

Proposition 2.10. Assume that v P C0pp´T , 0s, C2pΩqqXC1pp´T , 0s, C1pΩqq satisfies

the inequality (2.16) and either homogeneous Dirichlet or homogeneous Neumann boundary conditions. Suppose also that v vanishes to infinite order in space at px0, 0q, for some x0 P Ω. Then, vpx, 0q vanishes for any x P Ω and therefore vpx, tq

identically vanishes in Ω ˆ r´T, 0s.

3

The local infinite-dimensional dynamical

sys-tem S

f

ptq

In this section, we recall some basic properties of the local dynamical system Sfptq

generated by the parabolic equation (1.1) on Xα (if the dependence on f is clear,

we simply write Sptq). As we have seen in the introduction, the hyperbolicity of the critical elements (that is, the equilibrium points and periodic orbits) and the transversality of the stable and unstable manifolds play a primordial role. Thus, we will focus on recalling the definitions and main properties of these objects.

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3.1

Critical elements and hyperbolicity

Let e P Xα be an equilibrium point of (1.1). The linearization pD

uSptqeq of the

dynamical system Sptq at e is given by the linear semigroup eLet on Xα, where

Le: Dp∆Dq ÞÑ LppΩq is the linear operator defined by

Lev “ ∆Dv ` fu1px, epxq, ∇epxqqv ` f∇u1 px, epxq, ∇epxqq.∇v .

The operator ´Le is a sectorial operator and a Fredholm operator with compact

resolvent. Therefore, the spectrum of Leconsists of a sequence of isolated eigenvalues

of finite multiplicity, the norms of which converge to infinity. Since the resolvent of Le : X Ñ X is compact, the linear C0-semigroup eLet from X into X is compact

and its spectrum consists of a sequence of isolated eigenvalues of finite multiplicity converging to 0. By [49, Chapter 2, Theorem 2.4], µ is an eigenvalue of eLe if and

only if µ “ eλ, where λ is an eigenvalue of L e.

Definition 3.1. The equilibrium point e is said simple if 1 does not belong to the spectrum of eLe. The equilibrium point e is hyperbolic if eLe has no spectrum on

the unit circle S1

” tz P C | |z| “ 1u.

In the case of the equation (1.1), we may equivalently say that the equilibrium point e is simple if and only if 0 is not an eigenvalue of Le and that it is hyperbolic

if and only if Le has no eigenvalue with zero real part.

The Morse index ipeq is the (finite) number of eigenvalues of eLe of norm

strictly larger than 1 (counted with their multiplicities) or equivalently the number of eigenvalues of Le with positive real part.

Let pptq be a periodic solution of the scalar parabolic equation (1.1) with period ω ą 0. This periodic solution describes the periodic orbit Γ “ tpptq | t P r0, ωqu. The linearization of the dynamical system Sptq along pptq is given by the evolution operator Πf,ppt, sq : vs P Xα ÞÑ vptq P Xα, t ě s, where vpτ q solves the

non-autonomous equation "

Bτvpx, τ q “ ∆vpx, τ q ` fu1px, p, ∇pqvpx, τ q ` f∇u1 px, p, ∇pq∇vpx, τ q

vpx, sq “ vspxq .

(3.1) The operator Πf,ppω, 0q is called the (corresponding) period map. One remarks

that Πf,ppt ` ω, tq “ Πf,ppt ` mω, t ` pm ´ 1qωq for any t ě 0 and any m P N. Notice

that Btpptq is a solution of (3.1) and thus that 1 is an eigenvalue of Πf,ppω, 0q with

eigenvector Btpp0q. We emphasize that, due to the smoothing properties in finite

positive time of the parabolic equation (3.1), the operator Πf,ppt, sq : Xα Ñ Xα,

t ą s, is compact. Therefore, the spectrum of Πf,ppt ` ω, tq consists of a sequence

of isolated eigenvalues of finite multiplicity, converging to 0. As for the linearized operator eLe at the equilibrium point e, 0 is the only point where the spectrum of

Πf,ppt ` ω, tq accumulates. Actually, by the backward uniqueness property, 0 is not

an eigenvalue neither of eLe, nor of Π

f,ppt`ω, tq. By [31, Lemma 7.2.2], the spectrum

σpΠf,ppt ` ω, tqq of Πf,ppt ` ω, tq is independent of t P r0, `8q. For this reason, the

following definition makes sense.

To simplify the notation, when there is no confusion, we will simply write Πpt, sq instead of Πf,ppt, sq.

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Definition 3.2. A periodic solution pptq of period ω is simple or non-degenerate if the number 1 is a simple (isolated) eigenvalue of Πf,ppω, 0q.

The periodic solution pptq is hyperbolic if Πf,ppω, 0q has no spectrum on the

unit circle S1 except the eigenvalue one, which is simple and isolated.

Since Πf,ppω, 0q is a compact operator, the periodic solution pptq is hyperbolic if

and only if 1 is a simple, isolated eigenvalue of Πf,ppω, 0q and is the only eigenvalue

on the unit circle.

The Morse index ippq of pp¨q, or the Morse index ipΓq of Γ, is the (finite) number of eigenvalues of Πf,ppω, 0q of norm strictly larger than 1 (counted with their

multiplicities).

In what follows, we will sometimes say that the periodic orbit Γ “ tpptq | t P r0, ωqu is simple (resp. hyperbolic), instead of saying that pptq is simple (resp. hyperbolic).

A first important consequence of the simplicity property is the persistence of equilibrium points and periodic orbits under perturbations.

Theorem 3.3. Let r ě 2 be given and let f0 P Cr.

1) Let e0 be a simple equilibrium point of (1.1) with f “ f0. There exist a

neigh-borhood N of f0 in Cr and a neighborhood U of e0 in Xα such that, for any f P N ,

there exists a unique equilibrium point epf q in U . This equilibrium depends con-tinuously on f P Cr. In addition, the eigenvalues of L

epf q continuously depend on

f P Cr.

Moreover, if e0 is hyperbolic, the neighborhoods N and U can be chosen small

enough so that epf q is also hyperbolic and so that the Morse index ipeq is equal to ipe0q.

2) Let p0ptq be a simple periodic solution with period (resp. minimal period) ω0 of

(1.1) for f “ f0. There exist a neighborhood N of f0 in Cr, a positive number

η and a neighborhood U of Γ0 “ tp0ptq | t P r0, ω0qu in Xα such that, for any

f P N , there exists a unique periodic orbit Γpf q “ tppf qptq | t P r0, ωpf qqu in U , of period (resp. minimal period) ωpf q with |ωpf q ´ ω0| ď η. The period ωpf q and

the periodic orbit Γpf q continuously depend on f . In addition, the eigenvalues of Πf,ppf qpωpf q, 0q continuously depend on f P Cr.

Moreover, if f0 is hyperbolic, the neighborhoods N and U and η ą 0 can be chosen

small enough so that the periodic solution ppf qptq is hyperbolic and so that the Morse index ipppf qq is equal to the Morse-index ipp0q.

Proof: The first statement about the persistence of simple equilibria e0 is very

classical. Assume that }e0}L8 ď m and }∇e0}L8 ď m. Then, applying the implicit

function theorem or the fixed point theorem of strict contraction (see the proof [7, Lemma 4.c.2]), one shows that there exist a neighborhood N0 of f0 in CrpΩ ˆ

r´2m, 2ms ˆ r´2m, 2msdq and a neighborhood U of e0 in Xα such that for any

f P N0, there exists a unique equilibrium point epf q in U . This equilibrium depends

continuously of f P N0 and, moreover, all the other properties of the first statement

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a neighborhood N of f0 in Cr such that, for any f P N , there exists a unique

equilibrium point epf q in U and that all the other properties of the first statement hold.

Let p0ptq be a simple periodic solution of period ω0 ą 0 of (1.1) for f “ f0.

As-sume that suptPr0,ω0q}p0ptq}L8 ď m and suptPr0,ω

0q}∇p0ptq}L8 ď m. The statement

of the persistence of a simple periodic solution pfptq near p0ptq with period ωf close

to ω0and also of the uniqueness (up to a time translation) of this periodic solution, if

f belongs to a small enough neighborhood of f0 in CrpΩ ˆ r´2m, 2ms ˆ r´2m, 2msdq,

is a direct consequence of [31, Theorem 8.3.2]; it is proved by using the method of Poincar´e sections and the implicit function theorem or the fixed point theorem of strict contraction (for further results in the case where the perturbations are less regular, see also [23] and [24]). One concludes like in the proof of the statement 1) by using the restriction mapping R of Section 2.1.

The continuous dependence of the eigenvalues of Lepf q or of Πf,ppf qpωpf q, 0q with

respect to f P Cr is a consequence of the proof of the continuity results of Kato

(see [40, Theorems IX.24, IV.31, IV.3.18]) and of the properties of the restriction mapping R. Detailed proofs of continuity of the point spectrum can also be found

in [22, Section 3]. ˝

Notice that a periodic solution pptq of period ω can be simple, whereas the same periodic solution pptq, considered as periodic solution of period nω can be non-simple. This is the case when the spectrum of Πf,ppω, 0q contains a n-th root of 1.

Thus, in the statement 2) of Theorem 3.3, when p0ptq is a simple periodic solution

of period ω0 of (1.1) for f “ f0, we do not know if Γpf q “ tppf qptq | t P r0, ωpf qqu

is the unique periodic orbit of (1.1) in the neighborhood U of Γ0 if f belongs to

N . Indeed,if the spectrum of Πf0,p0pω0, 0q contains a n-th root of unity, then it is

possible that new periodic orbits of period close to nω0 are created (in the case

where n “ 2, it is the famous “period-doubling bifurcation”).

Of course, when p0ptq is hyperbolic, no such new periodic solutions can be created

and Γpf q is still isolated in the set of periodic orbits. Hyperbolicity is a notion independent of the chosen period.

3.2

Stable and unstable manifolds

We recall that a critical element means either an equilibrium point or a periodic orbit of (1.1).

Definition 3.4. Let C be a critical element of (1.1). The global stable and unstable sets of C are respectively defined as

WspCq “ tu0 P Xα| Sfptqu0 ÝÝÝÝÑ tÑ`8 Cu ,

WupCq “ tu0 P Xα| @ t ď 0, Sfptqu0 is well defined and Sfptqu0 ÝÝÝÝÑ tÑ´8 Cu .

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unstable sets of C defined as

WspC, UCq ” Wlocs pCq ” tu0 P UC| Sfptqu0 P UC, t ě 0u ,

WupC, UCq ” Wlocu pCq ” tu0 P UC| @ t ď 0, Sfptqu0 is well defined and stays in UCu.

If we need to specify the dependence with respect to the non-linearity f , we will denote these manifolds as Ws

pC, UC, f q and WupC, UC, f q or as Wlocs pC, f q and

Wu

locpC, f q.

Let e0 be an equilibrium point of (1.1) and let pDuSptqe0q “ eLe0t be the

corre-sponding linearized operator around e0. We denote by Pu (resp. Ps) the projection

in Xα onto the space generated by the (generalized) eigenfunctions of eLe0

corre-sponding to the eigenvalues with modulus strictly larger than 1 (resp. with modulus strictly smaller than 1). Let Xα

u “ PupXαq and Xsα “ PspXαq. We have seen that,

in the case of the parabolic equation (1.1), the Morse index of every hyperbolic equilibrium point is finite, which implies that PupXq “ PupXαq.

The following theorem states the existence of the local stable and unstable man-ifolds near hyperbolic equilibrium points. The result is very classical. In the case of a vector field on a finite-dimensional compact manifold, we refer the reader to [1], [47], [35] for example, and in the infinite dimensional case, we refer to [31], [26], [25], [11], [59].

Theorem 3.5. Let f0 be given in Cr, r ě 2, and let e0 be a hyperbolic equilibrium

point of Sf0ptq. Then there is a neighborhood U0 of e0 such that the local unstable

manifold Wupe

0, U0q (resp. the local stable manifold Wspe0, U0q) is a Cr-submanifold

of dimension ipe0q (resp. codimension ipe0q), which is tangent to Xuα (resp. Xsα) at

e0.

More precisely, there exist a neighborhood U0 of e0 in Xα, two mappings hupf0q ”

h0u : PuXα Ñ PsXα and hspf0q ” h0s : PsXα Ñ PuXαof class Cr such that h0up0q “ 0,

Dh0 up0q “ 0, h0sp0q “ 0, Dh0sp0q “ 0 and Wlocu pe0, f0q ” Wupe0, U0, f0q “ tv P U0| v “ e0` Pupv ´ e0q ` h0upPupv ´ e0qqu Wlocs pe0, f0q ” Wspe0, U0, f0q “ tv P U0| v “ e0` Pspv ´ e0q ` h0spPspv ´ e0qqu . (3.2)

Furthermore, the convergence rates to the origin are exponential. More precisely, there are positive constants k1, k2 and constants 0 ă γ2 ă 1 ă γ1, such that,

}Sf0ptqx}X ď k1γ t 1, @ t ď 0 , @ x P W u pe0, U0q , }Sf0ptqx}X ď k2γ t 2, @ t ě 0 , @ x P Wspe0, U0q . (3.3)

In addition, the local stable and unstable manifolds “continuously” depend of the nonlinear map f . More precisely, there exists ρ ą 0 and, for any ε ą 0, there is a neighborhood N of f0 in Cr such that, for any f P N , Sfptq has a unique equilibrium

point epf q in the ball BXαpe0, ρq of center e0 and radius ρ in Xα, and }epf q´e0}Xα ď

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given by

Wlocu pepf q, f q ” Wupepf q, U0, f q

“ tv P U0| v “ epf q ` Pupv ´ epf qq ` hupf qpPupv ´ epf qqqu

Wlocs pepf q, f q ” Wlocs pepf q, U0, f q

“ tv P U0| v “ epf q ` Pspv ´ epf qq ` hspf qpPspv ´ epf qqqu ,

where hupf q : PuXα Ñ PsXα and hspf q : PsXα Ñ PuXα are maps of class Cr such

that hupf qp0q “ 0, hspf qp0q “ 0 and }hupf q ´ h0u}Cr ď ε and }hspf q ´ h0s}Cr ď ε.

Finally, for any f P N , the above constants ki, γi are independent of f .

Proof: We refer to [31, Theorems 5.2.1. and 5.2.2] for the existence of the local sta-ble and unstasta-ble manifolds in the case of a hyperbolic equilibrium point of a parabolic equation. To obtain the last part of the Theorem, that is the smooth dependence with respect to f , we simply use a fixed point theorem with parameter. Indeed, the proof of Theorem 5.2.1 of [31] consists in constructing the mappings hu and hs as

fixed points of suitable contraction mappings. These maps depend smoothly on f and thus remain contractions mappings for f close to f0 and their fixed points hupf q

and hspf q depend smoothly on f . Notice that in general Dhupf qp0q and Dhspf qp0q

do not vanish, but are only small of order ε. ˝

Let ppx, tq be a hyperbolic periodic solution of (1.1) of minimal period ω ą 0, let Γ “ tpptq | t P r0, ωqu be the associated orbit and let Πpt, 0q : Xα Ñ Xα, be the

associated evolution operator defined by the linearized equation (3.1). We denote µi, i P N, the eigenvalues of the period map Πpω, 0q. Since ppx, tq is a hyperbolic

periodic solution, the intersection of the spectrum of Πpω, 0q with the unit circle S1 of C reduces to the eigenvalue 1, which is a simple (isolated) eigenvalue. We recall that, if ppaq, a P r0, ωq, is another point of the periodic orbit, the spectrum of DupSfpω, 0qppaqq coincides with the one of Πpω, 0q whereas the corresponding

eigenfunctions depend on the point ppaq.

We denote Pupaq (resp. Pcpaq, resp. Pspaq) the projection in Xα onto the space

generated by the (generalized) eigenfunctions of DupSfpω, 0qppaqq corresponding to

the eigenvalues with modulus strictly larger than 1 (resp. equal to 1, resp. with modulus strictly smaller than 1).

Since a hyperbolic periodic orbit is a particular case of a normally hyperbolic C1 manifold, we may apply, for example, the existence results of [5], [34], [35] or

[59, Theorem 14.2 and Remark 14.3] and thus, we may state the following theorem. Other methods of proofs are also given in [1], [35], [26], [25] and [47].

Theorem 3.6. Let f0 be given in Cr, r ě 2, and let Γ0 “ tp0ptq | t P r0, ω0qu be a

hyperbolic periodic orbit of Eq. (1.1) of minimal period ω0 ą 0.

1) There exists a small neighborhood UΓ0 of Γ0 in X

α such that the local unstable

and stable sets

Wlocu pΓ0q ” WupΓ0, UΓ0q “ tu0 P X α | Sf0ptqu0 P UΓ0, @t ď 0u Wlocs pΓ0q ” WspΓ0, UΓ0q “ tu0 P X α | Sf0ptqu0 P UΓ0, @t ě 0u

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are (embedded) C1-submanifolds of Xα of dimension ipΓ

0q ` 1 and codimension

ipΓ0q respectively.

2) Moreover, Wlocs pΓ0q and Wlocu pΓ0q are fibrated by the local strongly stable (resp.

unstable) manifolds at each point p0paq P Γ0, that is,

Wlocs pΓ0q “ YaPr0,ω0qW

ss

locpp0paqq , Wlocu pΓ0q “ YaPr0,ω0qW

su

locpp0paqq ,

where there exist positive constants ˜r0, κ0 and κ˚0 such that

Wlocsspp0paqq “tu0 P Xα| }Sf0ptqu0´ p0pa ` tq}Xα ă ˜r0 , @t ě 0 ,

lim

tÑ8e κ0t

}Sf0ptqu0´ p0pa ` tq}Xα “ 0u ,

Wlocsupp0paqq “tu0 P Xα| }Sf0ptqu0´ p0pa ` tq}Xα ă ˜r0 , @t ď 0 ,

lim tÑ´8e κ˚ 0t}S f0ptqu0´ p0pa ` tq}Xα “ 0u . (3.4)

For any a P r0, ω0q, Wlocsupp0paqq (resp. Wlocsspp0paqq) is a Cr-submanifold of Xα

of dimension ipΓq (resp. of codimension ipΓq ` 1) tangent at p0paq to PupaqXα

(resp. PspaqXα).

3) Finally, the local stable and unstable manifolds of the periodic orbit continuously depend on the nonlinear map f P Cr.

We have seen that the local stable and unstable manifolds are Cr graphs over

PsXα and PuXα respectively. In general, the global stable and unstable manifolds

are not embedded submanifolds of Xα.

Adapting the proof of [31, Theorem 6.1.9], one easily shows the following result. Theorem 3.7. Let f P Cr, r ě 2, be given.

1) Let e0 be a hyperbolic equilibrium point of (1.1). Then, the global unstable

set Wupe0q (resp. global stable set Wspe0q) is an injectively immersed invariant

manifold of class Cr in Xα of dimension (resp. of codimension) ipe 0q.

2) Likewise, let Γ0 “ tp0ptq | t P r0, ω0su be a hyperbolic periodic orbit of

mini-mal period ω0 ą 0. Then, the global unstable set WupΓ0q (resp. global stable

set WspΓ0q) is an injectively immersed invariant manifold of class Cr in Xα of

dimension ipΓ0q ` 1 (resp. of codimension ipΓ0q).

Proof: We will give the proof in the case of a hyperbolic equilibrium e0, since the

proof is very similar in the case of a hyperbolic periodic orbit.

Proof for the unstable manifold: For every m P N, we introduce the open set U0pmq “ tx P U0| Sfptqx is well defined, 0 ď t ď mu ,

where U0 is the neighborhood of e0, in which the local stable and unstable manifolds

are given as graphs (see Theorem 3.5). By Proposition 2.1, U0pmq is an open subset

of U0 and thus Wlocu pe0q X U0pmq is an open subset of Wlocu pe0q. We readily check

that

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Moreover, since Wu

locpe0q is negatively invariant, we have, for any m P N,

SfpmqpWlocu pe0q X U0pmqq Ă Sfpm ` 1qpWlocu pe0q X U0pm ` 1qq .

By Corollary 2.6, Sfpmq is an injective map from U0pmq into Xα. Moreover, by

Proposition 2.5, for any x P U0pmq, DuSfptqx is an injective map from Xα into

itself, thus Sfpmq|U0pmq is an injective C

r-immersion. By Theorem 3.5, Wu

locpe0q is

the image of an injective Cr-map H

u from the open ball BRkp0, 1q of center 0 and

radius 1 of Rk into Xα, where k “ ipe0q. Moreover, the derivative DHupyq has rank

k at each point y P BRkp0, 1q. We recall that Hu´1pWlocu pe0q X U0pmqq is an open

subset V pk, mq of BRkp0, 1q. It follows that SfpmqWlocu pe0q X U0pmqq is the image of

the injective Cr-immersion Sfpmq ˝ Hu : V pk, mq Ñ Xαand thus is a Cr-submanifold

of dimension k. Since the invariance is obvious, Statement 1) is proved. Proof for the stable manifold: We first remark that

Wspe0q “ Y`8m“0Sfpmq´1pWlocs pe0qq . (3.6)

Moreover, since Wlocs pe0q is positively invariant, we have, for any m P N,

Sfpmq´1pWlocs pe0qq Ă Sfpm ` 1q´1pWlocs pe0qq .

As a consequence of the property (3.2) in Theorem 3.5, where h0sis a Cr-map of PsXα

into the k-dimensional space PuXα and where Dh0sp0q “ 0, Wlocs pe0q is actually

represented as the set tv P U0| gpvq “ 0u, where g : x P U0 ÞÑ gpxq P Rk is a

map of class Cr and Dgpvq has constant rank k at every point v P g´1p0q. By

[31, Theorem 7.3.3], DSfpmqu has dense range at every point u P Xα at which

Sfpmqu exists if pDSfpmquq˚ is injective. By Proposition 2.5, the adjoint equation

(2.14) also satisfies the backward uniqueness property. Thus DSfpmqu has dense

range at every point u P Sfp´mqWlocs pe0q, which implies that, at every point u P

pg ˝ Sfpmqq´1p0q, DpgpSfpmqquq has rank k. In other terms, the mapping v Ñ

gpSfpmqvq is a submersion of constant rank k at every point u P pg ˝ Sfpmqq´1p0q.

By a theorem on Page 12 of [44] for example, pg ˝ Sfpmqq´1p0q is a Cr-submanifold

of Xα of codimension k. Thus, since S

f pmq is injective, Wspe0q is an injectively

immersed manifold of codimension k. Since the invariance is obvious, Statement 2)

is proved. ˝

3.3

Transversality of connecting orbits

We use here the above concepts of stable and unstable manifolds of hyperbolic equilibrium points or periodic orbits. The definitions related to Theorem 1.1 are as follows.

Definition 3.8. Let C˘ be two hyperbolic critical elements. We say that Wu

pC´q and Ws

pC`q intersect transversally (or are transverse) and we denote it by WupC´q&WspC`q ,

if, at each intersection point u0 P WupC´q X WspC`q, Tu0W

upC´q splits, that is,

contains a closed complement of Tu0W

s

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It is important to notice that, in this paper, the complement of Tu0W

s in Xα is

always closed since Tu0W

u

pC´q is finite-dimensional. Also note that, by definition, manifolds which do not intersect are transverse.

Definition 3.9. Let C´ ‰ C` be two different hyperbolic critical elements. A

trajectory uptq of Sptq is a heteroclinic orbit connecting C´ to C` if uptq P

Wu

pC´q X WspC`q.

Let C be a hyperbolic critical element. A trajectory uptq of Sptq is a homoclinic orbit to C if uptq P WupCq X WspCq.

A heteroclinic or homoclinic orbit is transverse if the above intersections of stable and unstable manifolds are transverse.

4

Singular nodal sets for linear parabolic

equa-tions with parameter

In this section, we consider a general linear parabolic equation with parameter Btvpx, t, τ q “ ∆vpx, t, τ q ` apx, t, τ qvpx, t, τ q ` bpx, t, τ q.∇xvpx, t, τ q , (4.1)

in a domain Ω of Rd.

We are interested in the singular nodal set of v, that is the points px, t, τ q where v and ∇xv both vanish. To this end, we use techniques coming from [29]. The singular

nodal set of solutions of the parabolic equations, with coefficients independent of the parameter τ , has already been studied in [28] and in [10]. Notice that we assume that v is smooth in the variables px, tq P Ω ˆ R, but this is not a restriction since this property holds in the applications, that we have in mind (see Section 5). Theorem 4.1. Let I and J be open intervals of R. Let a P C8

pΩ ˆ I ˆ J, Rq and b P C8

pΩ ˆ I ˆ J, Rdq be bounded coefficients. Let v be a strong solution of (4.1) with Dirichlet boundary conditions. Let r ě 1 and assume that v is of class Cr with

respect to τ and of class C8 with respect to x and t. Assume moreover that there

are no time t P I and no parameter τ P J such that vp., t, τ q ” 0. Then,

1) M “ tpx, t, τ q P Ω ˆ I ˆ J | vpx, t, τ q “ 0 , ∇xvpx, t, τ q “ 0u is contained in a

countable union of Cr

´manifolds of dimension d,

• either parametrized by t, τ and d ´ 2 components of x, • or parametrized by τ and d ´ 1 components of x. 2) the set

pT N Sq “ tpx0, t0q P Ω ˆ I | Eτ P J such that pvpx0, t0, τ q, ∇vpx0, t0, τ qq “ p0, 0qu

is generic in Ω ˆ I.

Proof: We introduce the set

Mq “ tpx, t, τ q PΩ ˆ I ˆ J such that for all |α| ď q , Dαxvpx, t, τ q “ 0,

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By Proposition 2.10, if vpx, t, τ q vanishes at infinite order in x, then vp., t, τ q identi-cally vanishes in Ω. By assumption, this is precluded. Thus, M “ Yqě1Mq. And,

without loss of generality, we can replace M by Mq in Property 1) of Theorem 4.1.

Let q ě 1 and px0, t0, τ0q P Ω ˆ I ˆ J . Let us first prove that there exists ρ0,q ą 0

such that Property 1) of Theorem 4.1 holds with Ω ˆ I ˆ J replaced by the ball Bppx0, t0, τ0q, ρ0,qq and M replaced by Mq. Assume that px0, t0, τ0q P Mq (otherwise

the property is trivial). There exists a multi-index β with |β| “ q ´ 1 such that HesspDβxvpx0, t0, τ0qq ‰ 0. In particular, there exist i, j, 1 ď i, j ď d, such that

the derivative D2 xixjpD

βvpx

0, t0, τ0qq ‰ 0. We next consider the Dxβ derivative of the

equation (4.1). Since v vanishes at order |β| ` 1 at px0, t0, τ0q, we obtain the equality

d dtD

βvpx

0, t0, τ0q “ ∆xpDβvpx0, t0, τ0qq .

Now two cases can occur: • Either d dtD βvpx 0, t0, τ0q “ 0 and thus řd k“1 B2 Bx2kpD βvpx 0, t0, τ0qq “ 0. In this case, if B2 Bx2kpD βvpx

0, t0, τ0qq “ 0 for all k, then there exist i ‰ j, such that

D2 xixjpD

βvpx

0, t0, τ0qq ‰ 0. By considering their ith and jth components, we

see that ∇xDxipD

βvpx

0, t0, τ0qq and ∇xDxjpD

βvpx

0, t0, τ0qq are linearly

inde-pendent. If, on the contrary, there exists i such that B2 Bx2ipD

βvpx

0, t0, τ0qq ‰ 0,

then there also exists j ‰ i such that B2 Bx2i pDβvpx0, t0, τ0qq ˆ B2 Bx2j pDβvpx0, t0, τ0qq ă 0 .

By considering their ith and jth components, we notice again that the vectors

∇xDxipD

βvpx

0, t0, τ0qq and ∇xDxjpD

βvpx

0, t0, τ0qq are linearly independent.

To summarize, in all the cases, there exist i and j, such that the vectors ∇xDxipD

βvpx

0, t0, τ0qq and ∇xDxjpD

βvpx

0, t0, τ0qq are linearly independent.

This implies that there exists ρ0,qą 0 such that

Bppx0, t0, τ0q, ρ0,qq X pDxiD βvq´1 p0q X pDxjD βvq´1 p0q is an embedded Cr

´submanifold Mqpx0, t0, τ0q in Rd`2 of dimension d which

contains all of Bppx0, t0, τ0q, ρ0,qq X Mq. This submanifold can be written as

Mqpx0, t0, τ0q “ tpx, t, τ q P Bppx0, t0, τ0q, ρ0,qq such that

pxi, xjq “ pΦippxkqk‰i,j, t, τ q, Φjppxkqk‰i,j, t, τ qqu .

• Or d dtD

βvpx

0, t0, τ0q ‰ 0, then there exists i such that Dx2iD

βvpx

0, t0, τ0q ‰

0. Notice that, since DxiD

βvpx 0, t0, τ0q “ 0, pDxi, DtqD βvpx 0, t0, τ0q and pDxi, DtqpDxiD βvpx

0, t0, τ0qq are linearly independent. Thus, there exists ρ0,q ą

0 such that

Bppx0, t0, τ0q, ρ0,qq X pDxiD

βvq´1

p0q X pDβvq´1p0q is an embedded Cr

´submanifold Mqpx0, t0, τ0q in Rd`2 of dimension d, which

contains all of Bppx0, t0, τ0q, ρ0,qq X Mq. This submanifold can be written as

Mqpx0, t0, τ0q “ px, t, τ q P Bppx0, t0, τ0q, ρ0,qq such that

Figure

Figure 1: A typical transversal heteroclinic orbit connecting a periodic orbit C ´ and an equilibrium point C `
Figure 2: A figure illustrating the proof of Proposition 6.2. In the phase space, N ˘ are small enough to define local dynamics and are disjoints in the heteroclinic case.
Figure 3: The geometric idea behind Sard-Smale theorems as Theorem B.3: if per- per-turbing the parameter λ provides enough freedom, a non-transversal intersection between ΦpM, λq and W is generically perturbed into either an empty, and thus transversal, i

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